Difference between revisions of "2025 AMC 8 Problems/Problem 17"
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| − | + | ==Problem== | |
| + | In the land of Markovia, there are three cities: \( A \), \( B \), and \( C \). There are 100 people who live in \( A \), 120 who live in \( B \), and 160 who live in \( C \). Everyone works in one of the three cities, and a person may work in the same city where they live. In the figure below, an arrow pointing from one city to another is labeled with the fraction of people living in the first city who work in the second city. (For example, \( \frac{1}{4} \) of the people who live in \( A \) work in \( B \).) How many people work in \( A \)? | ||
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| + | <math>\textbf{(A)}\ 55\qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 85\qquad \textbf{(D)}\ 115\qquad \textbf{(E)}\ 160</math> | ||
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| + | ==Solution 1== | ||
| + | <math>100 (1/4 + 1/5) = 100 \cdot \frac{9}{20} = 45</math> people do not work in city <math>A</math> that live in city <math>A</math>, meaning <math>55</math> people that live in city <math>A</math> work in city <math>A</math>. <math>\frac{1}{3} \cdot 120 = 40</math> people that live in <math>B</math> work in <math>A</math> and <math>\frac{1}{8} \cdot 160 = 20</math> people that live in <math>C</math> work in <math>A</math>, so the answer is <math>55 + 40 + 20 = \boxed{\textbf{(D)}115}</math>. | ||
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| + | ~ alwaysgonnagiveyouup | ||
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| + | ==Video Solution by Thinking Feet== | ||
| + | https://youtu.be/PKMpTS6b988 | ||
| + | |||
| + | ==See Also== | ||
| + | {{AMC8 box|year=2025|num-b=16|num-a=18}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 19:14, 30 January 2025
Problem
In the land of Markovia, there are three cities: \( A \), \( B \), and \( C \). There are 100 people who live in \( A \), 120 who live in \( B \), and 160 who live in \( C \). Everyone works in one of the three cities, and a person may work in the same city where they live. In the figure below, an arrow pointing from one city to another is labeled with the fraction of people living in the first city who work in the second city. (For example, \( \frac{1}{4} \) of the people who live in \( A \) work in \( B \).) How many people work in \( A \)?
Solution 1
people do not work in city 
that live in city , meaning 
people that live in city work in city . 
people that live in 
work in and 
people that live in 
work in , so the answer is 
.
~ alwaysgonnagiveyouup
Video Solution by Thinking Feet
See Also
| 2025 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
